metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.26Q16, C28.50SD16, C28.8M4(2), C42.195D14, Q8⋊(C7⋊C8), C7⋊3(Q8⋊C8), (C7×Q8)⋊1C8, C28.8(C2×C8), C14.15C4≀C2, (C4×Q8).1D7, C4⋊C4.4Dic7, (Q8×C28).1C2, (Q8×C14).7C4, (C2×C28).490D4, C4.16(Q8⋊D7), (C2×Q8).4Dic7, C28⋊C8.11C2, (C4×C28).46C22, C4.14(C7⋊Q16), C4.2(C4.Dic7), C14.16(C22⋊C8), C2.2(Q8⋊Dic7), C14.10(Q8⋊C4), C2.3(D4⋊2Dic7), C2.6(C28.55D4), C22.30(C23.D7), C4.2(C2×C7⋊C8), (C4×C7⋊C8).4C2, (C7×C4⋊C4).6C4, (C2×C28).60(C2×C4), (C2×C4).38(C2×Dic7), (C2×C4).162(C7⋊D4), (C2×C14).93(C22⋊C4), SmallGroup(448,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.26Q16
G = < a,b,c | a28=b8=1, c2=a14b4, bab-1=a13, ac=ca, cbc-1=a21b-1 >
Subgroups: 196 in 70 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C7⋊C8, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8⋊C8, C2×C7⋊C8, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, Q8×C14, C4×C7⋊C8, C28⋊C8, Q8×C28, C28.26Q16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), SD16, Q16, Dic7, D14, C22⋊C8, Q8⋊C4, C4≀C2, C7⋊C8, C2×Dic7, C7⋊D4, Q8⋊C8, C2×C7⋊C8, C4.Dic7, Q8⋊D7, C7⋊Q16, C23.D7, C28.55D4, Q8⋊Dic7, D4⋊2Dic7, C28.26Q16
►Regular action on 448 points
Generators in S448
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308)(309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364)(365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392)(393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 50 112 440 364 234 318 149)(2 35 85 425 337 247 319 162)(3 48 86 438 338 232 320 147)(4 33 87 423 339 245 321 160)(5 46 88 436 340 230 322 145)(6 31 89 421 341 243 323 158)(7 44 90 434 342 228 324 143)(8 29 91 447 343 241 325 156)(9 42 92 432 344 226 326 141)(10 55 93 445 345 239 327 154)(11 40 94 430 346 252 328 167)(12 53 95 443 347 237 329 152)(13 38 96 428 348 250 330 165)(14 51 97 441 349 235 331 150)(15 36 98 426 350 248 332 163)(16 49 99 439 351 233 333 148)(17 34 100 424 352 246 334 161)(18 47 101 437 353 231 335 146)(19 32 102 422 354 244 336 159)(20 45 103 435 355 229 309 144)(21 30 104 448 356 242 310 157)(22 43 105 433 357 227 311 142)(23 56 106 446 358 240 312 155)(24 41 107 431 359 225 313 168)(25 54 108 444 360 238 314 153)(26 39 109 429 361 251 315 166)(27 52 110 442 362 236 316 151)(28 37 111 427 363 249 317 164)(57 269 299 203 415 169 379 116)(58 254 300 216 416 182 380 129)(59 267 301 201 417 195 381 114)(60 280 302 214 418 180 382 127)(61 265 303 199 419 193 383 140)(62 278 304 212 420 178 384 125)(63 263 305 197 393 191 385 138)(64 276 306 210 394 176 386 123)(65 261 307 223 395 189 387 136)(66 274 308 208 396 174 388 121)(67 259 281 221 397 187 389 134)(68 272 282 206 398 172 390 119)(69 257 283 219 399 185 391 132)(70 270 284 204 400 170 392 117)(71 255 285 217 401 183 365 130)(72 268 286 202 402 196 366 115)(73 253 287 215 403 181 367 128)(74 266 288 200 404 194 368 113)(75 279 289 213 405 179 369 126)(76 264 290 198 406 192 370 139)(77 277 291 211 407 177 371 124)(78 262 292 224 408 190 372 137)(79 275 293 209 409 175 373 122)(80 260 294 222 410 188 374 135)(81 273 295 207 411 173 375 120)(82 258 296 220 412 186 376 133)(83 271 297 205 413 171 377 118)(84 256 298 218 414 184 378 131)
(1 179 350 265)(2 180 351 266)(3 181 352 267)(4 182 353 268)(5 183 354 269)(6 184 355 270)(7 185 356 271)(8 186 357 272)(9 187 358 273)(10 188 359 274)(11 189 360 275)(12 190 361 276)(13 191 362 277)(14 192 363 278)(15 193 364 279)(16 194 337 280)(17 195 338 253)(18 196 339 254)(19 169 340 255)(20 170 341 256)(21 171 342 257)(22 172 343 258)(23 173 344 259)(24 174 345 260)(25 175 346 261)(26 176 347 262)(27 177 348 263)(28 178 349 264)(29 405 227 61)(30 406 228 62)(31 407 229 63)(32 408 230 64)(33 409 231 65)(34 410 232 66)(35 411 233 67)(36 412 234 68)(37 413 235 69)(38 414 236 70)(39 415 237 71)(40 416 238 72)(41 417 239 73)(42 418 240 74)(43 419 241 75)(44 420 242 76)(45 393 243 77)(46 394 244 78)(47 395 245 79)(48 396 246 80)(49 397 247 81)(50 398 248 82)(51 399 249 83)(52 400 250 84)(53 401 251 57)(54 402 252 58)(55 403 225 59)(56 404 226 60)(85 200 333 127)(86 201 334 128)(87 202 335 129)(88 203 336 130)(89 204 309 131)(90 205 310 132)(91 206 311 133)(92 207 312 134)(93 208 313 135)(94 209 314 136)(95 210 315 137)(96 211 316 138)(97 212 317 139)(98 213 318 140)(99 214 319 113)(100 215 320 114)(101 216 321 115)(102 217 322 116)(103 218 323 117)(104 219 324 118)(105 220 325 119)(106 221 326 120)(107 222 327 121)(108 223 328 122)(109 224 329 123)(110 197 330 124)(111 198 331 125)(112 199 332 126)(141 368 446 302)(142 369 447 303)(143 370 448 304)(144 371 421 305)(145 372 422 306)(146 373 423 307)(147 374 424 308)(148 375 425 281)(149 376 426 282)(150 377 427 283)(151 378 428 284)(152 379 429 285)(153 380 430 286)(154 381 431 287)(155 382 432 288)(156 383 433 289)(157 384 434 290)(158 385 435 291)(159 386 436 292)(160 387 437 293)(161 388 438 294)(162 389 439 295)(163 390 440 296)(164 391 441 297)(165 392 442 298)(166 365 443 299)(167 366 444 300)(168 367 445 301)
G:=sub<Sym(448)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,50,112,440,364,234,318,149)(2,35,85,425,337,247,319,162)(3,48,86,438,338,232,320,147)(4,33,87,423,339,245,321,160)(5,46,88,436,340,230,322,145)(6,31,89,421,341,243,323,158)(7,44,90,434,342,228,324,143)(8,29,91,447,343,241,325,156)(9,42,92,432,344,226,326,141)(10,55,93,445,345,239,327,154)(11,40,94,430,346,252,328,167)(12,53,95,443,347,237,329,152)(13,38,96,428,348,250,330,165)(14,51,97,441,349,235,331,150)(15,36,98,426,350,248,332,163)(16,49,99,439,351,233,333,148)(17,34,100,424,352,246,334,161)(18,47,101,437,353,231,335,146)(19,32,102,422,354,244,336,159)(20,45,103,435,355,229,309,144)(21,30,104,448,356,242,310,157)(22,43,105,433,357,227,311,142)(23,56,106,446,358,240,312,155)(24,41,107,431,359,225,313,168)(25,54,108,444,360,238,314,153)(26,39,109,429,361,251,315,166)(27,52,110,442,362,236,316,151)(28,37,111,427,363,249,317,164)(57,269,299,203,415,169,379,116)(58,254,300,216,416,182,380,129)(59,267,301,201,417,195,381,114)(60,280,302,214,418,180,382,127)(61,265,303,199,419,193,383,140)(62,278,304,212,420,178,384,125)(63,263,305,197,393,191,385,138)(64,276,306,210,394,176,386,123)(65,261,307,223,395,189,387,136)(66,274,308,208,396,174,388,121)(67,259,281,221,397,187,389,134)(68,272,282,206,398,172,390,119)(69,257,283,219,399,185,391,132)(70,270,284,204,400,170,392,117)(71,255,285,217,401,183,365,130)(72,268,286,202,402,196,366,115)(73,253,287,215,403,181,367,128)(74,266,288,200,404,194,368,113)(75,279,289,213,405,179,369,126)(76,264,290,198,406,192,370,139)(77,277,291,211,407,177,371,124)(78,262,292,224,408,190,372,137)(79,275,293,209,409,175,373,122)(80,260,294,222,410,188,374,135)(81,273,295,207,411,173,375,120)(82,258,296,220,412,186,376,133)(83,271,297,205,413,171,377,118)(84,256,298,218,414,184,378,131), (1,179,350,265)(2,180,351,266)(3,181,352,267)(4,182,353,268)(5,183,354,269)(6,184,355,270)(7,185,356,271)(8,186,357,272)(9,187,358,273)(10,188,359,274)(11,189,360,275)(12,190,361,276)(13,191,362,277)(14,192,363,278)(15,193,364,279)(16,194,337,280)(17,195,338,253)(18,196,339,254)(19,169,340,255)(20,170,341,256)(21,171,342,257)(22,172,343,258)(23,173,344,259)(24,174,345,260)(25,175,346,261)(26,176,347,262)(27,177,348,263)(28,178,349,264)(29,405,227,61)(30,406,228,62)(31,407,229,63)(32,408,230,64)(33,409,231,65)(34,410,232,66)(35,411,233,67)(36,412,234,68)(37,413,235,69)(38,414,236,70)(39,415,237,71)(40,416,238,72)(41,417,239,73)(42,418,240,74)(43,419,241,75)(44,420,242,76)(45,393,243,77)(46,394,244,78)(47,395,245,79)(48,396,246,80)(49,397,247,81)(50,398,248,82)(51,399,249,83)(52,400,250,84)(53,401,251,57)(54,402,252,58)(55,403,225,59)(56,404,226,60)(85,200,333,127)(86,201,334,128)(87,202,335,129)(88,203,336,130)(89,204,309,131)(90,205,310,132)(91,206,311,133)(92,207,312,134)(93,208,313,135)(94,209,314,136)(95,210,315,137)(96,211,316,138)(97,212,317,139)(98,213,318,140)(99,214,319,113)(100,215,320,114)(101,216,321,115)(102,217,322,116)(103,218,323,117)(104,219,324,118)(105,220,325,119)(106,221,326,120)(107,222,327,121)(108,223,328,122)(109,224,329,123)(110,197,330,124)(111,198,331,125)(112,199,332,126)(141,368,446,302)(142,369,447,303)(143,370,448,304)(144,371,421,305)(145,372,422,306)(146,373,423,307)(147,374,424,308)(148,375,425,281)(149,376,426,282)(150,377,427,283)(151,378,428,284)(152,379,429,285)(153,380,430,286)(154,381,431,287)(155,382,432,288)(156,383,433,289)(157,384,434,290)(158,385,435,291)(159,386,436,292)(160,387,437,293)(161,388,438,294)(162,389,439,295)(163,390,440,296)(164,391,441,297)(165,392,442,298)(166,365,443,299)(167,366,444,300)(168,367,445,301)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,50,112,440,364,234,318,149)(2,35,85,425,337,247,319,162)(3,48,86,438,338,232,320,147)(4,33,87,423,339,245,321,160)(5,46,88,436,340,230,322,145)(6,31,89,421,341,243,323,158)(7,44,90,434,342,228,324,143)(8,29,91,447,343,241,325,156)(9,42,92,432,344,226,326,141)(10,55,93,445,345,239,327,154)(11,40,94,430,346,252,328,167)(12,53,95,443,347,237,329,152)(13,38,96,428,348,250,330,165)(14,51,97,441,349,235,331,150)(15,36,98,426,350,248,332,163)(16,49,99,439,351,233,333,148)(17,34,100,424,352,246,334,161)(18,47,101,437,353,231,335,146)(19,32,102,422,354,244,336,159)(20,45,103,435,355,229,309,144)(21,30,104,448,356,242,310,157)(22,43,105,433,357,227,311,142)(23,56,106,446,358,240,312,155)(24,41,107,431,359,225,313,168)(25,54,108,444,360,238,314,153)(26,39,109,429,361,251,315,166)(27,52,110,442,362,236,316,151)(28,37,111,427,363,249,317,164)(57,269,299,203,415,169,379,116)(58,254,300,216,416,182,380,129)(59,267,301,201,417,195,381,114)(60,280,302,214,418,180,382,127)(61,265,303,199,419,193,383,140)(62,278,304,212,420,178,384,125)(63,263,305,197,393,191,385,138)(64,276,306,210,394,176,386,123)(65,261,307,223,395,189,387,136)(66,274,308,208,396,174,388,121)(67,259,281,221,397,187,389,134)(68,272,282,206,398,172,390,119)(69,257,283,219,399,185,391,132)(70,270,284,204,400,170,392,117)(71,255,285,217,401,183,365,130)(72,268,286,202,402,196,366,115)(73,253,287,215,403,181,367,128)(74,266,288,200,404,194,368,113)(75,279,289,213,405,179,369,126)(76,264,290,198,406,192,370,139)(77,277,291,211,407,177,371,124)(78,262,292,224,408,190,372,137)(79,275,293,209,409,175,373,122)(80,260,294,222,410,188,374,135)(81,273,295,207,411,173,375,120)(82,258,296,220,412,186,376,133)(83,271,297,205,413,171,377,118)(84,256,298,218,414,184,378,131), (1,179,350,265)(2,180,351,266)(3,181,352,267)(4,182,353,268)(5,183,354,269)(6,184,355,270)(7,185,356,271)(8,186,357,272)(9,187,358,273)(10,188,359,274)(11,189,360,275)(12,190,361,276)(13,191,362,277)(14,192,363,278)(15,193,364,279)(16,194,337,280)(17,195,338,253)(18,196,339,254)(19,169,340,255)(20,170,341,256)(21,171,342,257)(22,172,343,258)(23,173,344,259)(24,174,345,260)(25,175,346,261)(26,176,347,262)(27,177,348,263)(28,178,349,264)(29,405,227,61)(30,406,228,62)(31,407,229,63)(32,408,230,64)(33,409,231,65)(34,410,232,66)(35,411,233,67)(36,412,234,68)(37,413,235,69)(38,414,236,70)(39,415,237,71)(40,416,238,72)(41,417,239,73)(42,418,240,74)(43,419,241,75)(44,420,242,76)(45,393,243,77)(46,394,244,78)(47,395,245,79)(48,396,246,80)(49,397,247,81)(50,398,248,82)(51,399,249,83)(52,400,250,84)(53,401,251,57)(54,402,252,58)(55,403,225,59)(56,404,226,60)(85,200,333,127)(86,201,334,128)(87,202,335,129)(88,203,336,130)(89,204,309,131)(90,205,310,132)(91,206,311,133)(92,207,312,134)(93,208,313,135)(94,209,314,136)(95,210,315,137)(96,211,316,138)(97,212,317,139)(98,213,318,140)(99,214,319,113)(100,215,320,114)(101,216,321,115)(102,217,322,116)(103,218,323,117)(104,219,324,118)(105,220,325,119)(106,221,326,120)(107,222,327,121)(108,223,328,122)(109,224,329,123)(110,197,330,124)(111,198,331,125)(112,199,332,126)(141,368,446,302)(142,369,447,303)(143,370,448,304)(144,371,421,305)(145,372,422,306)(146,373,423,307)(147,374,424,308)(148,375,425,281)(149,376,426,282)(150,377,427,283)(151,378,428,284)(152,379,429,285)(153,380,430,286)(154,381,431,287)(155,382,432,288)(156,383,433,289)(157,384,434,290)(158,385,435,291)(159,386,436,292)(160,387,437,293)(161,388,438,294)(162,389,439,295)(163,390,440,296)(164,391,441,297)(165,392,442,298)(166,365,443,299)(167,366,444,300)(168,367,445,301) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308),(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336),(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364),(365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392),(393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,50,112,440,364,234,318,149),(2,35,85,425,337,247,319,162),(3,48,86,438,338,232,320,147),(4,33,87,423,339,245,321,160),(5,46,88,436,340,230,322,145),(6,31,89,421,341,243,323,158),(7,44,90,434,342,228,324,143),(8,29,91,447,343,241,325,156),(9,42,92,432,344,226,326,141),(10,55,93,445,345,239,327,154),(11,40,94,430,346,252,328,167),(12,53,95,443,347,237,329,152),(13,38,96,428,348,250,330,165),(14,51,97,441,349,235,331,150),(15,36,98,426,350,248,332,163),(16,49,99,439,351,233,333,148),(17,34,100,424,352,246,334,161),(18,47,101,437,353,231,335,146),(19,32,102,422,354,244,336,159),(20,45,103,435,355,229,309,144),(21,30,104,448,356,242,310,157),(22,43,105,433,357,227,311,142),(23,56,106,446,358,240,312,155),(24,41,107,431,359,225,313,168),(25,54,108,444,360,238,314,153),(26,39,109,429,361,251,315,166),(27,52,110,442,362,236,316,151),(28,37,111,427,363,249,317,164),(57,269,299,203,415,169,379,116),(58,254,300,216,416,182,380,129),(59,267,301,201,417,195,381,114),(60,280,302,214,418,180,382,127),(61,265,303,199,419,193,383,140),(62,278,304,212,420,178,384,125),(63,263,305,197,393,191,385,138),(64,276,306,210,394,176,386,123),(65,261,307,223,395,189,387,136),(66,274,308,208,396,174,388,121),(67,259,281,221,397,187,389,134),(68,272,282,206,398,172,390,119),(69,257,283,219,399,185,391,132),(70,270,284,204,400,170,392,117),(71,255,285,217,401,183,365,130),(72,268,286,202,402,196,366,115),(73,253,287,215,403,181,367,128),(74,266,288,200,404,194,368,113),(75,279,289,213,405,179,369,126),(76,264,290,198,406,192,370,139),(77,277,291,211,407,177,371,124),(78,262,292,224,408,190,372,137),(79,275,293,209,409,175,373,122),(80,260,294,222,410,188,374,135),(81,273,295,207,411,173,375,120),(82,258,296,220,412,186,376,133),(83,271,297,205,413,171,377,118),(84,256,298,218,414,184,378,131)], [(1,179,350,265),(2,180,351,266),(3,181,352,267),(4,182,353,268),(5,183,354,269),(6,184,355,270),(7,185,356,271),(8,186,357,272),(9,187,358,273),(10,188,359,274),(11,189,360,275),(12,190,361,276),(13,191,362,277),(14,192,363,278),(15,193,364,279),(16,194,337,280),(17,195,338,253),(18,196,339,254),(19,169,340,255),(20,170,341,256),(21,171,342,257),(22,172,343,258),(23,173,344,259),(24,174,345,260),(25,175,346,261),(26,176,347,262),(27,177,348,263),(28,178,349,264),(29,405,227,61),(30,406,228,62),(31,407,229,63),(32,408,230,64),(33,409,231,65),(34,410,232,66),(35,411,233,67),(36,412,234,68),(37,413,235,69),(38,414,236,70),(39,415,237,71),(40,416,238,72),(41,417,239,73),(42,418,240,74),(43,419,241,75),(44,420,242,76),(45,393,243,77),(46,394,244,78),(47,395,245,79),(48,396,246,80),(49,397,247,81),(50,398,248,82),(51,399,249,83),(52,400,250,84),(53,401,251,57),(54,402,252,58),(55,403,225,59),(56,404,226,60),(85,200,333,127),(86,201,334,128),(87,202,335,129),(88,203,336,130),(89,204,309,131),(90,205,310,132),(91,206,311,133),(92,207,312,134),(93,208,313,135),(94,209,314,136),(95,210,315,137),(96,211,316,138),(97,212,317,139),(98,213,318,140),(99,214,319,113),(100,215,320,114),(101,216,321,115),(102,217,322,116),(103,218,323,117),(104,219,324,118),(105,220,325,119),(106,221,326,120),(107,222,327,121),(108,223,328,122),(109,224,329,123),(110,197,330,124),(111,198,331,125),(112,199,332,126),(141,368,446,302),(142,369,447,303),(143,370,448,304),(144,371,421,305),(145,372,422,306),(146,373,423,307),(147,374,424,308),(148,375,425,281),(149,376,426,282),(150,377,427,283),(151,378,428,284),(152,379,429,285),(153,380,430,286),(154,381,431,287),(155,382,432,288),(156,383,433,289),(157,384,434,290),(158,385,435,291),(159,386,436,292),(160,387,437,293),(161,388,438,294),(162,389,439,295),(163,390,440,296),(164,391,441,297),(165,392,442,298),(166,365,443,299),(167,366,444,300),(168,367,445,301)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | SD16 | Q16 | D14 | Dic7 | Dic7 | C4≀C2 | C7⋊D4 | C7⋊C8 | C4.Dic7 | Q8⋊D7 | C7⋊Q16 | D4⋊2Dic7 |
| kernel | C28.26Q16 | C4×C7⋊C8 | C28⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | C7×Q8 | C2×C28 | C4×Q8 | C28 | C28 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C2×C4 | Q8 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 12 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of C28.26Q16 ►in GL5(𝔽113)
| 98 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 80 | 34 |
| 0 | 0 | 0 | 71 | 9 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 105 | 8 | 0 | 0 |
| 0 | 105 | 105 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 104 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 25 | 74 | 0 | 0 |
| 0 | 74 | 88 | 0 | 0 |
| 0 | 0 | 0 | 34 | 101 |
| 0 | 0 | 0 | 68 | 79 |
G:=sub<GL(5,GF(113))| [98,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,80,71,0,0,0,34,9],[18,0,0,0,0,0,105,105,0,0,0,8,105,0,0,0,0,0,0,104,0,0,0,25,0],[1,0,0,0,0,0,25,74,0,0,0,74,88,0,0,0,0,0,34,68,0,0,0,101,79] >;
C28.26Q16 in GAP, Magma, Sage, TeX
C_{28}._{26}Q_{16} % in TeX
G:=Group("C28.26Q16"); // GroupNames label
G:=SmallGroup(448,92);
// by ID
G=gap.SmallGroup(448,92);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,100,1123,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14*b^4,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^21*b^-1>;
// generators/relations